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Negative Komar Mass of Single Objects in Regular, Asymptotically LETTER TO THE EDITOR Negative Komar Mass of Single Objects in Regular, Asymptotically Flat Spacetimes M Ansorg1 and D Petroff2 1 Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany 2 Theoretisch-Physikalisches Institut, University of Jena, Max-Wien-Platz 1, 07743 Jena, Germany E-mail: [email protected], [email protected] Abstract. We study two types of axially symmetric, stationary and asymptotically flat spacetimes using highly accurate numerical methods. The one type contains a black hole surrounded by a perfect fluid ring and the other a rigidly rotating disc of dust surrounded by such a ring. Both types of spacetime are regular everywhere (outside of the horizon in the case of the black hole) and fulfil the requirements of the positive energy theorem. However, it is shown that both the black hole and the disc can have negative Komar mass. Furthermore, there exists a continuous transition from discs to black holes even when their Komar masses are negative. PACS numbers: 04.70.Bw, 04.40.-b, 04.25.Dm preprint number: AEI-2006-057 A great many definitions for mass in General Relativity have been proposed. Many of them consider only the spacetime as a whole, whereas others assign a mass locally to a portion of the spacetime. Even the latter are not always applicable to a single object in a spacetime containing multiple ones. In [1], Bardeen considers an axially symmetric, stationary spacetime containing a black hole surrounded by a perfect fluid matter distribution and assigns to each of arXiv:gr-qc/0607091v2 15 Nov 2006 the two objects a mass based on the Komar integral [2]. In a similar way, he assigns an angular momemtum to each of the objects. Various other local mass definitions include the Hawking mass [3], the Bartnik mass [4], the Christodoulou mass [5], and a related mass used for isolated horizons, which can be generalized to dynamical ones [6, 7]. For a review article see [8]. A mass definition applicable both to matter and black holes, and which can be used for single components of a many-body problem in stationary spacetimes, is the one based on the Komar integral. We will refer to it here as the Komar mass even when applied to single objects. In this letter, we study axially symmetric and stationary spacetimes containing either a black hole or a rigidly rotating disc of dust, each surrounded by a perfect fluid ring. We find that there exists a continuous transition from the disc to the black hole. If one is interested in being able to talk about the mass of the individual objects in such stationary spacetimes and since the transition from the disc to the black hole exists, then one is led to look for a definition that is applicable in either scenario. Letter to the Editor 2 We thus choose to examine the behaviour of the Komar mass and find that it can become negative both for the black hole and for the disc although the total mass of the spacetime is of course positive as guaranteed‡ by the positive energy theorem. We find morover that the continuous transition mentioned above exists even when the Komar mass of each of the two central objects is negative. The Poisson equation in Newtonian gravity 2U =4πε (1) ∇ relates the potential U to the mass density ε . The mass contained in any volume of space can be defined by integrating Eq. (1)§ over that volume and applying the divergenceV theorem: 1 M = ε d3x = U dF . (2) ZV 4π I∂V ∇ · Obviously a region of space containing no matter has zero mass and the total mass can be found using Mtot = lim (rU). − r→∞ Moreover, mass is additive, i.e. the mass contained in a region of space is simply the sum of the mass contained in each subregion of an arbitrary subdivision of that space. A similar procedure can be used to define a relativistic mass in axially symmetric and stationary spacetimes. Such a spacetime containing black holes and perfect fluids with strictly azimuthal motion can be described in Weyl-Lewis-Papapetrou coordinates by the line element 2 ds2 = e2µ(d̺2 + dζ2)+ ̺2B2e−2ν (dϕ ω dt) e2ν dt2, − − where the metric functions depend only on ̺ and ζ. The energy-momentum tensor of the perfect fluid is T µν = (ε + p)uµuν + pgµν , where ε is the energy density of the fluid, p its pressure and uµ its four-velocity. The Komar integral for the mass can be constructed by integrating Einstein’s equation Rt =8π T t 1 T t t − 2 over any volume in the 3-space generated by taking t = constant. Applying the divergence theorem again then yields (see [1]) 1 1 M = ε˜ d3x = B ν ̺2B3e−4ν ω ω dF , (3) ZV 4π I∂V ∇ − 2 ∇ · with 1+ v2 v ε˜ := e2µB (ε + p) +2p +2̺Be−2ν (ε + p) ω 1 v2 1 v2 − − v := ̺Be−2ν (Ω ω), − where Ω is the angular velocity of a fluid element with respect to infinity and the vector operators have the same meaning as in a Euclidean space in which (̺, ϕ, ζ) are cylindrical coordinates. ‡ Negative horizon masses have also been observed for rotating black holes of Einstein-Maxwell- Chern-Simons theory [9]. § We use units in which the gravitational constant and speed of light are equal to one, G = c = 1. Letter to the Editor 3 The mass defined in the surface integral of Eq. (3) is the Komar mass (we use the term here even when is of finite extent). One can see that here too a regular volume containing no matterV (i.e. ε = p = 0) has zero mass. If one is considering a black hole spacetime, then the region interior to the horizon can be excised and the surface integral in Eq. (3) used to define the mass of the black hole. The mass of any single object in a stationary spacetime could thus be defined by calculating the surface integral in Eq. (3) over a surface containing that object and only that object. It can be used both for matter and for black holes and has the additive property familiar from Newtonain theory that the sum of the masses of the single objects equals the total (ADM) mass of the spacetime Mtot = lim (rν). − r→∞ A quantity, which plays an important role in black hole thermodynamics and is constant over the horizon is the surface gravity. In coordinates, such as the ones chosen in this letter, in which the horizon is a sphere, it reads ∂ κ = e−µ eν , r := ̺2 + ζ2. ∂r p Smarr [10] showed that for the Komar mass of the black hole κA Mh = + 2ΩhJh (4) 4π always holds, true even in the presence of a surrounding ring [1], see also [11]. Here Ωh is the angular velocity of the black hole (i.e. the constant value of the function ω over the horizon), Jh its angular momentum and A its area. A similar expression can be derived for the rigidly rotating disc of dust (cf. III.15 in [12]), also valid in the presence of a surrounding ring. It turns out that the potential ′ gtt in a coordinate system co-rotating with the disc (and denoted here by the prime) must be a constant along its ‘surface’ ′ 2 2 2 d g = e ν 1 v =: e V0 . (5) − tt − d This constant is related to the relative redshift Z0 of photons with zero angular momentum emitted from the surface of the disc and observed at infinity via the equation d d −V0 Z0 = e 1. − Making use of the definition for the baryonic mass t 3 M0 = εu √ g d x, Z − we can write the disc’s Komar mass as d V Md = e 0 M0 + 2ΩdJd. (6) The first term on the right hand side of Eqs (4) and (6) is non-negative. In the absence of the ring, the angular velocity and angular momentum must have the same sign, so that the second term is also non-negative. If however a ring is present, it can induce a frame-dragging effect which allows Ω and J of the central object to have d different signs. If moreover κ (or eV0 ) becomes sufficiently small, then the expression for the Komar mass could become negative. Our results demonstrate that this indeed occurs. Letter to the Editor 4 In order to study spacetimes containing a black hole surrounded by a rigidly rotating ring with constant energy density, we make use of the multi-domain pseudo- spectral method described in [13]. To study the scenario containing a disc as opposed to a black hole, we made appropriate modifications to the program. In the upper plot of Fig. 1 we consider sequences of configurations that demonstrate the existence of negative Komar masses for discs and black holes surrounded by rings. In the absence of a ring, the analytic solution for the disc is known [14] and depends on one ‘physical’ and one scaling parameter as does the Kerr metric. When a ring is present, the configurations are described by four physical parameters, so that a sequence can be specified by holding three parameters constant and varying a fourth. Along the sequences in the plot, the ratio of proper inner to outer circumference of the ring was held at a value of Ci/Co =0.85 and a mass-shed parameter for the outer edge of the ring was held at a value βo = 0.3. For the disc sequence,k the ratio of its coordinate radius to that of the outer edge of the ring was chosen to be ̺d/̺o = 0.1, whereas for the black hole sequence rh/̺o = 0.1 was chosen.
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